Theoretical Study of the Trapped Dipolar Bose Gas in the Ultra-Cold Regime

The work of this thesis concerns the properties of a Bose gas of polarised dipoles that have long-range and anisotropic interactions. Our work is divided into two parts. First, the stability of a dipolar Bose gas at finite temperature (both above and below the critical Bose-Einstein condensation (BEC) temperature Tc). Second, the fluctuations of a dipolar BEC at zero and small finite temperature (T ≪ Tc) in a regime where rotonic excitations emerge. Part I: Stability of a Trapped Finite Temperature Dipolar Bose Gas: Above Tc we implement a semiclassical Hartree-Fock theory and characterise the dependence of the stability boundary on temperature, trap geometry and the strength of the dipole-dipole interaction and contact interaction. We find that stability is greatly enhanced above Tc and that trap geometry continues to play a key role. Furthermore, we find that for oblate traps a novel double instability feature emerges. To extend our stability analysis to the low temperature regime, T < Tc, we develop a beyond semiclassical Hartree theory. We use this to characterise the stability boundary as a function of geometry. Interestingly, we find large beyond semiclassical effects above Tc for prolate trapping geometries. We characterise thermal effects on biconcave condensate states. Part II: Rotons and Fluctuations in a Trapped Dipolar Condensate: To study density fluctuations we implement a numerical scheme to solve the Gross-Pitaevskii equation and the Bogoliubov de Gennes equations. We find that the phonon and roton gases spatially separate and we characterise the role of the anomalous density on the density fluctuations of the thermally activated rotons. We develop a numerical scheme that calculates number fluctuations within cells of various shapes and sizes, and find a strong peak in the fluctuations when the cell size is around half the roton wavelength, which should be detectable by current experiments. By tailoring the cell shape we predict that experiments should be able to detect the effects of individual roton modes. For the study of zero temperature fluctuations we deploy the Gross-Pitaevskii and Bogoliubov de Gennes equations to calculate the dynamic and static structure factors for a highly oblate BEC. We find a clear signature of the roton gas dispersion relation within the structure factors. This signature should be detectible in current experiments using Bragg spectroscopy